Dynamical Origin of the Electroweak Scale and the 125 GeV Scalar

# Dynamical Origin of the Electroweak Scale and the 125 GeV Scalar

Stefano Di Chiara    Roshan Foadi Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014, University of Jyväskylä, Finland Helsinki Institute of Physics, P.O. Box 64, FI-000140, University of Helsinki, Finland    Kimmo Tuominen Department of Physics, University of Helsinki, P.O. Box 64, FI-000140, University of Helsinki, Finland Helsinki Institute of Physics, P.O. Box 64, FI-000140, University of Helsinki, Finland    Sara Tähtinen Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014, University of Jyväskylä, Finland Helsinki Institute of Physics, P.O. Box 64, FI-000140, University of Helsinki, Finland
###### Abstract

We consider a fully dynamical origin for the masses of weak gauge bosons and heavy quarks of the Standard Model. Electroweak symmetry breaking and the gauge boson masses arise from new strong dynamics, which leads to the appearance of a composite scalar in the spectrum of excitations. In order to generate mass for the Standard Model fermions, we consider extended gauge dynamics, effectively represented by four fermion interactions at presently accessible energies. By systematically treating these interactions, we show that they lead to a large reduction of the mass of the scalar resonance. Therefore, interpreting the scalar as the recently observed 125 GeV state, implies that the mass originating solely from new strong dynamics can be much heavier, i.e. of the order of 1 TeV. In addition to reducing the mass of the scalar resonance, we show that the four-fermion interactions allow for contributions to the oblique corrections in agreement with the experimental constraints. The couplings of the scalar resonance with the Standard Model gauge bosons and fermions are evaluated, and found to be compatible with the current LHC results. Additional new resonances are expected to be heavy, with masses of the order of a few TeVs, and hence accessible in future experiments.

preprint: HIP-2014-39/TH

## I Introduction

The discovery of the Higgs boson at the Large Hadron Collider (LHC) Aad et al. (2012); Chatrchyan et al. (2012) established the Standard Model (SM) as an accurate description of elementary particle interactions Giardino et al. (2014); Azatov, Contino, and Galloway (2012); Alanne, Di Chiara, and Tuominen (2014). However, it is know that the SM is incomplete: For example, the SM itself does not provide any clue towards understanding the generational structure and mass patterns of the matter fields. Furthermore, understanding the origin of dark matter or the baryon-antibaryon asymmetry continue to provide motivation for searches of viable beyond-the-Standard-Model (BSM) scenarios.

So far the LHC has shown no sign of new particles typically predicted by various BSM setups, such as Technicolor (TC) and its variants (see Hill and Simmons (2003); Sannino (2009) for review). Furthermore, the lightest resonance in a model of dynamical electroweak symmetry breaking is naturally expected to be much heavier than GeV Foadi et al. (2007). These premature concerns rest on treating new strong dynamics in isolation, i.e. without taking the interaction with the SM fields into account. It is known that a light scalar can arise from approximate global symmetries, as in models where the Higgs is a pseudo Goldstone boson associated with chiral symmetry Kaplan and Georgi (1984); Kaplan, Georgi, and Dimopoulos (1984); Cacciapaglia and Sannino (2014) or scale invariance Yamawaki, Bando, and Matumoto (1986); Bando, Matumoto, and Yamawaki (1986); Sannino and Tuominen (2005); Dietrich, Sannino, and Tuominen (2005); Chacko and Mishra (2013). Only recently it has been realised that also with QCD-like TC dynamics the scalar particle can become light because of loop corrections originating from extended sectors, which are always required in TC models to account for the generation of fermion masses. In Foadi et al. (2007) a preliminary analysis, using simply SM-like Yukawa couplings to parametrize the effects from the coupling with the top quark, was carried out to point out this effect. In Di Chiara, Foadi, and Tuominen (2014) this effect was investigated in a fully dynamical model setup of simple extended technicolor (ETC). Within this model, a computation in the large- limit was carried out, where is the dimension of the technifermion representation under the TC gauge group. It was then possible to rigorously demonstrate a large reduction of the scalar mass from the value arising solely from new strong dynamics. The amount of fine tuning involved is on the tolerable level of a few per cent Di Chiara, Foadi, and Tuominen (2014). However, the model considered in Di Chiara, Foadi, and Tuominen (2014) was simple and devised only to illustrate this effect, and it could not be used for a realistic description of the origin of all mass scales of the SM.

In this paper we present a necessary further development of the model framework described above. We use a chiral fermion model, similar to the Nambu–Jona-Lasinio model (NLJ), to account for TC dynamics, and augment it with a whole set of four-fermion operators, low-energy remnants of ETC interactions. We show that the mechanism featured in Di Chiara, Foadi, and Tuominen (2014) for the reduction of the scalar mass also works in this case, and that the effective couplings of the composite Higgs particle with the SM particles are very close to the SM-Higgs couplings, and hence compatible with the LHC data111See also Espriu, Mescia, and Yencho (2013) for a related study.. We also compute the oblique corrections and demonstrate the viability of the model with respect to the electroweak precision data. One of our robust and generic findings within this framework is that in order to reduce the Higgs mass from values near 1 TeV, natural for new strong dynamics, to 125 GeV, the ETC interactions must be strongly coupled. However, we only consider scenarios in which the ETC interactions, although strong, are not strong enough to generate fermion condensation. Therefore, we complement the analysis of Lane (2014), where a model with strong ETC dynamics and weak TC interactions was considered.

Model building of the full gauge dynamics required by ETC theories is challenging Appelquist, Piai, and Shrock (2004). Our effective theory, formulated in terms of four fermion couplings, and taking into account only the third generation quarks, can hopefully be seen as a stepping stone towards more complete dynamical theories of flavour. There exists lots of earlier work using NJL-like models to describe dynamical electroweak symmetry breaking and the associated Higgs physics, see e.g.. Miransky, Tanabashi, and Yamawaki (1989a, b); Bardeen, Hill, and Lindner (1990); Dobrescu and Hill (1998); Chivukula et al. (1999); He, Hill, and Tait (2002); Fukano and Tuominen (2012, 2013); Geller, Bar-Shalom, and Soni (2014). There is also a large lattice program motivated by applications to BSM physics and aimed at studying strong dynamics in isolation Hietanen et al. (2009); Hietanen, Rummukainen, and Tuominen (2009); Karavirta et al. (2012); Appelquist et al. (2011, 2014); DeGrand, Shamir, and Svetitsky (2013); Fodor et al. (2012, 2014); Aoki et al. (2013, 2014). Our analysis should be applicable in refining the phenomenological interpretation of the lattice results.

The paper is organised as follows. In sections II and III we introduce the effective description of the strong TC dynamics, and the interactions arising from the ETC theory, in terms of a chiral-techniquark model augmented with four fermion interactions. In section. IV we show how confinement and cutoff are realised in the model when smearing the momentum integrals with a mass distribution density for the techniquarks. In section V we demonstrate how the fundamental SM fields acquire mass dynamically, whereas in section VI we prove that a strongly-coupled yet subcritical ETC theory may lead to a large reduction of the mass of the lightest scalar resonance from values near 1 TeV to 125 GeV. In section VII we compute the coupling of the scalar resonance with the fundamental SM fields, and in section VIII we evaluate the oblique electroweak paramaters. In section IX we present the numerical results of our analysis for two different TC theories, and compare with precision data as well as LHC results. Finally, in section X we conclude and discuss the further prospects.

## Ii Chiral-techniquark Lagrangian

We focus on TC theories featuring one colorless weak technidoublet, , in the complex -dimensional representation of the TC gauge group. We assume that there are no additional weak doublets. Therefore, in order to avoid the topological Witten anomaly, must be an even number. Since the spinorial representation is not complex, we must have . Cancellation of the standard gauge anomalies, and requiring the electromagnetic gauge group to remain unbroken, impose the hypercharge assignments

 YQL=0 ,YUR=12 ,YDR=−12 . (1)

In the limit of zero electroweak gauge couplings, the techniquark kinetic terms feature a global chiral symmetry, which is dynamically broken by the TC force to . The lightest states, and the only ones that we include in our analysis, are therefore expected to be the massless technipion triplet – which upon electroweak gauging become the longitudinal component of the and boson – and a scalar singlet , which will be identified with the Higgs particle. In order to model TC dynamics, we employ a chiral-techniquark Lagrangian featuring both constituent techniquarks and resonances. This reads

 L = L¯¯¯¯¯¯¯SM+¯¯¯¯QLi⧸DQL+¯¯¯¯URi⧸DUR+¯¯¯¯¯DRi⧸DDR−MQ(1+yMQH+⋯)(¯¯¯¯QLΣQR+¯¯¯¯QRΣ†QL) (2) − M22H2+⋯+LETC ,

where is the SM Lagrangian without the terms containing the Higgs doublet, the covariant derivatives are with respect to the electroweak gauge fields, the ellipses denote higher-order terms in , and the TC gauge indices have been suppressed from the techniquark fields. The field is the standard non-linear sigma-model field,

 Σ≡exp2iΠiTiv , (3)

where is the technipion triplet, are the Pauli matrices, and is the vacuum expectation value. The composite nature of the and fields in (2) is manifest because the corresponding kinetic terms are absent. These are generated radiatively, and vanish at some large compositeness scale. Finally, contain four-fermion operators which are obtained by integrating out the heavy ETC gauge bosons. These are considered in more detail in the next section.

## Iii Four-fermion operators from ETC

In we only consider four-fermion operators containing the techniquark doublet and the top-bottom doublet . In fact, operators built out of lighter SM fermions are expected to arise from exchanges of very heavy ETC bosons, and are therefore highly suppressed at the electroweak scale. We focus on ETC theories in which left-handed and right-handed fields belong to different representations, and classify the four-fermion operators according to the quantum numbers of the exchanged ETC gauge bosons. We assume that there is only one ETC gauge boson with a given set of quantum numbers, and that, under the TC and QCD gauge groups, the ETC bosons are either singlets or - and - multiplets, respectively. This categorizes the ETC bosons into five distinct classes which we call A, B, C, D, and E: The classes A and B correspond to bosons which are TC and QCD singlet with hypercharge (for class A) and (for class B). The classes C, D and E consist of bosons which are - and -multiplets of TC and QCD, respectively, with hypercharge for class C, for class D and for class E.

Below the ETC scale the ETC gauge bosons are heavy with masses . Integrating out the heavy bosons leads to effective four fermion interactions. Generally, the relevant terms in the fundamental Lagrangian are of the form

 LGETC∼gXX′¯¯¯¯¯XγμX′Gμ+M2GGμGμ∗, (4)

where and are any of the fermions , , , , , , and all interaction terms allowed by the representation of the ETC boson under consideration should be taken into account. Integrating out the ETC boson at tree-level gives first

 G∗μ∼−gXX′M2G¯¯¯¯¯XγμX′ , (5)

and, after plugging back in , one obtains the effective four fermion interaction

 LGETC∼−|gXX′|2M2G∣∣¯¯¯¯¯XγμX′∣∣2 , (6)

valid below the ETC scale. For example, for the class D and E bosons this procedure leads to

 LDETC=−|gUb|2M2D(¯¯¯¯URγμbR)(¯¯bRγμUR),LEETC=−|gDt|2M2E(¯¯¯¯¯DRγμtR)(¯tRγμDR) , (7)

where the quark color index has been suppressed. The complete results of this classification are given in Appendix A. Also, it is convenient to Fierz rearrange some of the products of fermion bilinears. The formulas which we use are given in Appendix B.

Note that the diagonal couplings are real, but the off-diagonal couplings , with , can be complex. We assume that also these couplings are real,

 g∗XY=gXY , (8)

i.e. we assume that there are no new sources of violation. Furthermore, for simplicity we assume that all ETC masses are identical:

 MA=MB=MC=MD=ME≡M . (9)

Putting together all ETC operators from Appendix A and using the Fierz rearrangement formulas from Appendix B, under the above assumptions, gives the ETC Lagrangian

 LETC = 2GQqUt[(¯¯¯¯QLUR)(¯tRqL)+(¯¯¯qLtR)(¯¯¯¯URQL)]+2GQqDb[(¯¯¯¯QLDR)(¯¯bRqL)+(¯¯¯qLbR)(¯DRQL)] (10) + 2GQQUU(¯¯¯¯QLUR)(¯¯¯¯URQL)+2GQQDD(¯¯¯¯QLDR)(¯¯¯¯¯DRQL)+2Gqqtt(¯¯¯qLtR)(¯tRqL)+2Gqqbb(¯¯¯qLbR)(¯¯bRqL) + ΔLETC ,

where the couplings are defined as

 GQqUt≡gQqgUtM2 ,GQqDb≡gQqgDbM2 , GQQUU≡gQQgUUNM2 ,GQQDD≡gQQgDDNM2 ,Gqqtt≡gqqgttNcM2 ,Gqqbb≡gqqgbbNcM2 . (11)

The contribution is more complicated and reads

 \allowdisplaybreaksΔLETC =−12g2QQM2(¯¯¯¯QLγμQL)2−12g2qqM2(¯¯¯qLγμqL)2−12g2UUM2(¯¯¯¯URγμUR)2−12g2DDM2(¯¯¯¯¯DRγμDR)2 −12g2ttM2(¯tRγμtR)2−12g2bbM2(¯¯bRγμbR)2−gQQgqq+g2Qq/2M2(¯¯¯¯QLγμQL)(¯¯¯qLγμqL) −gQQgttM2(¯¯¯¯QLγμQL)(¯tRγμtR)−gQQgbbM2(¯¯¯¯QLγμQL)(¯¯bRγμbR)−gqqgUUM2(¯¯¯qLγμqL)(¯¯¯¯URγμUR) −gqqgDDM2(¯¯¯qLγμqL)(¯¯¯¯¯DRγμDR)−gUUgDDM2(¯¯¯¯URγμUR)(¯¯¯¯¯DRγμDR)−gUUgtt+g2UtM2(¯¯¯¯URγμUR)(¯tRγμtR) −gUUgbb+g2UbM2(¯¯¯¯URγμUR)(¯¯bRγμbR)−gDDgtt+g2DtM2(¯¯¯¯¯DRγμDR)(¯tRγμtR) −gDDgbb+g2DbM2(¯¯¯¯¯DRγμDR)(¯¯bRγμbR)−gttgbbM2(¯tRγμtR)(¯¯bRγμbR)−g2UDM2(¯¯¯¯URγμDR)(¯¯¯¯¯DRγμUR) −g2tbM2(¯tRγμbR)(¯¯bRγμtR)−gUDgtb+gUtgDbM2[(¯¯¯¯URγμDR)(¯¯bRγμtR)+(¯¯¯¯¯DRγμUR)(¯tRγμbR)] −2g2QqM2(¯¯¯¯QLγμTiQL)(¯¯¯qLγμTiqL)+4gQQgUUM2(¯¯¯¯QLTATCUR)(¯¯¯¯URTATCQL) +4gQQgDDM2(¯¯¯¯QLTATCDR)(¯¯¯¯¯DRTATCQL)+4gqqgttM2(¯¯¯qLTaQCDtR)(¯tRTaQCDqL) +4gqqgbbM2(¯¯¯qLTaQCDbR)(¯¯bRTaQCDqL) , (12)

where are the TC generators for the representation, and are the generators for the fundamental representation of . These matrices are normalized as

 Tr TiTj=12δij ,Tr TaQCDTbQCD=12δab ,Tr TATCTBTC=12δAB . (13)

As we shall see below, the operators not included in contribute both to scalar and fermion masses, whereas the operators included in contribute to neither.222Note that in TC theories with near-conformal dynamics, four-fermion operators with techniquark bilinears may be enhanced relative to operators with quark bilinears. In this paper we will not pursue such more model dependent questions, but treat all four fermion interactions appearing in (10).

We compute observables in the large- limit, with finite. For a consistent large- expansion, the ETC couplings must scale like . Therefore, the and couplings in (11) scale like , whereas the diagonal couplings and scale like , the extra factor of arising from the Fierz rearrangement of the class-A operators. It is clear that the operators in (11) contribute to mass, as they involve bilinears mixing left-handed and right-handed fermions. Therefore, we get mass contribution at leading order (LO) in from the and operators, and contributions at next-to-leading order (NLO) from the and operators. On the other hand, the operators contained in contribute to fermion and scalar masses neither to LO nor to NLO in the large- expansion. At LO this is evident from the presence of uncontracted matrices in separate loops. The NLO is zero either because of the appearance of products (as in the case of left-left bilinear products and the operators with the TC and QCD generators), or because it is manifestly absent (as in the case of left-right bilinear products mixing quarks and techniquarks). Hence, we can consistently compute masses to LO and NLO by only considering the operators in (11). However, NLO computations are rather complicated. In this paper, we find it more convenient to formally treat the and couplings as quantities scaling like , and compute all observables to LO in the large- expansion. The error is still NLO in , but this approach allows us to account for the important mass contribution from the class-A operators333This approach is similar to the one adopted in topcolor-assisted technicolor for treating the new hypercharge interactions..

## Iv Cutoff and confinement

There are two physical cutoffs in the model: , associated to TC dynamics, and the ETC scale . Therefore, we are naturally led to use a cutoff regulator for the standard loop integrals. It is not clear, though, which one of the two cutoffs should be used to evaluate the integrals. A possible approach consists in using the smaller mass scale, which we assume to be . This, however, would imply losing information from the dynamics occurring between and . Furthermore, it is well know that making the techniquark loop integrals finite with a sharp cutoff does not account for confinement, as the fermion propagators go on-shell for sufficiently large external momenta. A solution to both problems is provided by models of confinement. In the model of Efimov and Ivanov (1993), for instance, the interaction of external mesons is given by amplitudes of the form

 iT(q1,q2,…,qn−1)≡−∫d4k(2π)4∫dMQ2πiρ(MQ)TriΓ1i(⧸k−⧸q1+MQ)(k−q1)2−M2Q iΓ2i(⧸k−⧸q1−⧸q2+MQ)(k−q1−q2)2−M2Q⋯iΓn−1i(⧸k−⧸q1−⧸q2−⋯−⧸qn−1+MQ)(k−q1−q2−⋯−qn−1)2−M2QiΓni(⧸k+MQ)k2−M2Q , (14)

where are matrices in Dirac space, and are determined by the quantum numbers of the external mesons. Here the fermion mass is a complex variable which is integrated along a closed contour enclosing the external momenta. Confinement and convergence of the integrals are both guaranteed by taking the function to be holomorphic everywhere and decreasing faster than any polynomial for . Under these assumptions we may use the Cauchy integral formula to obtain

 ρ(z)=∫dMQ2πiρ(MQ)MQ−z=1κa(−z2/κ2)+1κ2zb(−z2/κ2) , (15)

where is an intrinsic mass scale of confinement, and faster than any polynomial for . The equation above gives

 a(−z2/κ2)=κ∫dMQ2πiρ(MQ)MQM2Q−z2 ,b(−z2/κ2)=κ2∫dMQ2πiρ(MQ)M2Q−z2 . (16)

Consider for instance the two-point function for two external scalar mesons, that is :

 iT(q)≡−∫d4k(2π)4∫dMQ2πiρ(MQ)Trii(⧸k−⧸q+MQ)(k−q1)2−M2Qii(⧸k+MQ)k2−M2Q , (17)

After combining the denominators, reducing the powers of in the numerator, Wick rotating, shifting to Euclidean momentum, and changing the integration variable to , we obtain

 iT(q)=i4π21κ2∫10dx∫∞0du[−2u2ddu−u]b(u/κ2−x(1−x)q2/κ2) . (18)

Integrating by parts, and using the hypothesis that decreases faster than any polynomial for , leads to the result

 T(q)=34π2[κ2B1(q2)+q2B0(q2)] , (19)

where

 B0(q2) ≡ ∫10dx∫∞0dξx(1−x)b(ξ−x(1−x)q2/κ2) , B1(q2) ≡ ∫10dx∫∞0dξ(ξ−x(1−x)q2/κ2)b(ξ−x(1−x)q2/κ2) . (20)

These functions are finite and, featuring no pole singularity, imply fermion confinement.

It is interesting to compute the three-meson and four-meson interactions at zero external momenta. For three external mesons we have to compute integrals like

 −∫d4k(2π)4∫dMQ2πiρ(MQ)TriΓ1i(⧸k+MQ)k2−M2QiΓ2i(⧸k+MQ)k2−M2QiΓ3i(⧸k+MQ)k2−M2Q

In general, this requires evaluating an integral of the form

 iI(c1,c3)≡∫d4k(2π)4∫dMQ2πiρ(MQ)c1MQk2+c3M3Q(k2−M2Q)3 , (21)

where the coefficients and depend on the matrices. Employing the same techniques leading to (19) gives

 I(c1,c3)=116π21κ∫∞0du[(c1+c3)u22d2du2+c3uddu]a(u/κ2) . (22)

Integrating by parts twice, and using the hypothesis that decreases faster than any polynomial for , leads to

 I(c1,c3)=c1κ16π2∫∞0dξa(ξ) . (23)

In the case of four external mesons at zero external momenta, the integrals to be computed are like

 −∫d4k(2π)4∫dMQ2πiρ(MQ)TriΓ1i(⧸k+MQ)k2−M2QiΓ2i(⧸k+MQ)k2−M2QiΓ3i(⧸k+MQ)k2−M2QiΓ4i(⧸k+MQ)k2−M2Q .

This requires evaluating an integral of the form

 iI(c0,c2,c4)≡−∫d4k(2π)4∫dMQ2πiρ(MQ)c0(k2)2+c2M2Qk2+c4M4Q(k2−M2Q)4 , (24)

which eventually gives

 I(c0,c2,c4)=c016π2∫∞0dξb(ξ) . (25)

The interesting aspect of (23) and (25) is that only the highest power of momentum contributes to the loop integral. This is important, as it preserves the special relation between form factors which is implied by the underlying chiral symmetry. In fact, using a sharp cutoff, rather than a confining function, the terms with the highest power of loop momentum, in the three-point and four-point vertices, correspond to the leading divergent logarithm, which preserves the underlying chiral symmetry Delbourgo and Scadron (1995).

If we use a distribution density to smear the integrals over techniquarks, we may cutoff the full theory at . The integrals over SM quarks are cutoff at , whereas the integrals over techniquarks are naturally finite. Clearly we must choose an appropriate function , and integrals are unavoidably more difficult to evaluate than the standard loop integrals, especially in the presence of isospin mass splitting. However in our analysis we are only interested in small external momenta, and thus we are not concerned with unphysical thresholds. Therefore, we make the approximation of using a sharp cutoff for the loop integrals over techniquark momenta, rather than a distribution density, while still cutting off the SM-fermion loop integrals at . This approach allows the dynamics between and to contribute to the low-energy observables, and at the same time preserves the symmetries of the Lagrangian Di Chiara, Foadi, and Tuominen (2014). In accordance with the above results, our prescription is the following:

1. Compute integrals over techniquarks with a cutoff , and integrals over ordinary quarks with a cutoff .

2. In evaluating interaction vertices, retain only the logarithmically divergent part of the integral.

3. Evaluate the integrals at zero external momenta.

We will need to evaluate fermion loops with external weak bosons, hence we must use a regulator preserving gauge invariance. Since we are using a cutoff, we find it convenient to employ the regularization prescription of Cynolter and Lendvai (2011), and require that the relation

 ∫d4lE(2π)4lEμlEν(l2E+m2)n+1=gμν2n∫d4lE(2π)41(l2E+m2)n (26)

is satisfied, for integrals in Euclidean space, for any . After this condition is imposed, integrals may be evaluated with a sharp cutoff. In Cynolter and Lendvai (2011) this prescription is shown to satisfy the Ward identities.

We end this section by enlisting the standard integrals used for computing the two-point functions. In accordance to the prescription above, we evaluate these at zero external momentum:

 IX ≡ i∫d4k(2π)41k2−M2X , JXY ≡ −i∫10dx∫d4l(2π)41(l2−xM2X−(1−x)M2Y)2 , KXY ≡ −i∫10dx∫d4l(2π)4x(l2−xM2X−(1−x)M2Y)2 , LXY ≡ −i∫10dx∫d4l(2π)4x(1−x)(l2−xM2X−(1−x)M2Y)2 . (27)

In order to evaluate the scalar wavefunction renormalization, as well as the parameter, we also need to consider at finite external momentum , take the derivative with respect to , and evaluate the resulting integral at . Since the latter has dimension , we find it convenient to define

 J′XX ≡ i3∫d4l(2π)4M2X(l2−M2X)3 . (28)

We provide explicit expressions for these integrals in Appendix C.

## V Mass of the fundamental particles

### v.1 Fermion masses

To LO in the large- expansion, the fermion masses are given by the diagrams of Fig. 1. These lead to the coupled equations

 MU = MQ+4NGQQUUMUIU+4NcGQqUtMtIt Mt = 4NGQqUtMUIU+4NcGqqttMtIt , (29)

and

 MD = MQ+4NGQQDDMDID+4NcGQqDbMbIb Mb = 4NGQqDbMDID+4NcGqqbbMbIb . (30)

Note that unlike the model of Di Chiara, Foadi, and Tuominen (2014), in which the only ETC operator was the one proportional to , now we have additional ETC contributions to mass. In particular, the isospin splitting may be softened or even set to zero by adjusting the and operators. This removes the major obstacle of Di Chiara, Foadi, and Tuominen (2014): There, in order to obtain a large reduction of the TC-Higgs mass, the value of had to be increased. This, in turn, made considerably heavier than , and the parameter unacceptably large. Now, instead, contributions from and have a double effect: they reduce the amount of isospin splitting, and, as we shall see, contribute to further reduce the TC-Higgs mass.

### v.2 Weak boson masses

We may compute the mass in terms of the fermion masses. In order to do so we must compute the corresponding vacuum-polarisation amplitude (VPA), which is required by gauge invariance to be transverse:

 ΠμνWW(q)=ΠWW(q2)(gμν−qμqνq2) . (31)

The expression for can be extracted from the part of the amplitude. Ignoring contributions from , which are suppressed by a factor of the order of 444The contribution to and may be ignored as long as the corresponding contribution to the parameter is within experimental bounds, which is anyway a strict requirement for viability., the only contribution to arises from the one-loop diagrams of Fig. 2 (top). In order to recover a fully transverse result, one needs to include an infinite chain of fermion loops, as well as tree-level Goldstone bosons exchanges. Using the fermion mass equations, we have verified in the simplified case that transversality is recovered. From the one-loop diagrams we obtain

 ΠWW(q2) = −2g2[NLUD+NcLtb]q2+g2[NM2UKUD+NM2DKDU+NcM2tKtb+NcM2bKbt] . (32)

Since the boson has a tree-level kinetic term, to leading-order in the weak coupling we may ignore the first term. Then is given by 555Clearly the contribution from is completely negligible: however we display it in order to show the contribution from all isospin components.

 M2W=g2[NM2UKUD+NM2DKDU+NcM2tKtb+NcM2bKbt] . (33)

We may rewrite this equation as

 1√2GF=4[NM2UKUD+NM2DKDU+NcM2tKtb+NcM2bKbt] , (34)

where is the Fermi constant, GeV. Equation (34) generalizes the Pagels-Stokar equation by taking into account the ETC contributions. Using this equation, together with the fermion mass equations (29) and (30), we can solve for , , , , , as a function of , , , , , , , and the experimental values of , , and .

We finally compute the mass of the boson. The result is

 M2Z=g2+g′22[NM2UJUU+NM2DJDD+NcM2tJtt+NcM2bJbb] . (35)

## Vi Mass of the TC Higgs

The TC-Higgs self-energy is given by the chain of diagrams shown in Fig. 3. Including the tree-level mass , we obtain

 (36)

where

 ISSXX≡⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩\vspace0.3cmISSXX1−NGQQXXISSXX,X=U,DISSXX1−NcGqqXXISSXX,X=t,b , (37)

and

 ISSXY